Lp-Regularized Least Squares (0<p<1) and Critical Path

Abstract: The least squares problem is formulated in terms of Lp quasi-norm regularization (0<p<1). Two formulations are considered: (i) an Lp-constrained optimization and (ii) an Lp-penalized (unconstrained) optimization. Due to the nonconvexity of the Lp quasi-norm, the solution paths of the regularized least squares problem are not ensured to be continuous. A critical path, which is a maximal continuous curve consisting of critical points, is therefore considered separately. The critical paths are piecewise smooth, as can be seen from the viewpoint of the variational method, and generally contain non-optimal points such as saddle points and local maxima as well as global/local minima. Along each critical path, the correspondence between the regularization parameters (which govern the 'strength' of regularization in the two formulations) is non-monotonic and, more specifically, it has multiplicity. Two paths of critical points connecting the origin and an ordinary least squares (OLS) solution are highlighted. One is a main path starting at an OLS solution, and the other is a greedy path starting at the origin. Part of the greedy path can be constructed with a generalized Minkowskian gradient. The breakpoints of the greedy path coincide with the step-by-step solutions generated by using orthogonal matching pursuit (OMP), thereby establishing a direct link between OMP and Lp-regularized least squares.